Math 5334: Differential Geometry

Course Description: This course covers both the differential and the geometric parts of differential geometry.  The differential part includes manifolds, the tangent bundle of a manifold, differential forms, tensors, and Riemannian metrics.  All of this is used in formulating the geometric part, which includes curvature and torsion of space curves, Gaussian curvature of surfaces, the Gauss-Bonnet Theorem, covariant derivatives, parallel transport, and the Riemann curvature tensor on manifolds of higher dimension.

List of Topics:

1. Manifolds

Examples of manifolds

Coordinate charts

Transition functions between coordinate charts

Atlases on manifolds

Differentiability of functions between manifolds

The Jacobian

The chain rule

2. The Tangent Bundle

Description of vectors as linear operators

Transformations of vectors under changes of coordinates

Vector fields

3. The Cotangent Bundle

Differential 1-forms

Transformations of differential forms under a change of coordinates

4. Tensors

Covariant and contravariant tensors

Raising and lowering of indices (via a Riemannian metric)

5. Inner Products

The matrix of an inner product with respect to a basis

Riemannian metrics on manifolds

6. Curves in the Plane and in Space

Curvature and torsion of plane curves and space curves

The Serret-Frenet formulas

7. Curvature of Surfaces in Space

Gaussian curvature

The first and second fundamental forms

The Theorema Egregium

8. Differential Geometry on Higher-dimensional Manifolds

Riemann curvature tensor (its definition and meaning)

Torsion tensor (its definition and meaning)

Covariant derivatives and the Levi-Civita connection

Parallel transport

9. The Gauss-Bonnet Theorem

References:

1. Michael Spivak, A Comprehensive Introduction to Differential Geometry, Volumes I and II (and Vol. III for the Gauss-Bonnet Theorem)

2. Manfredo Do Carmo, Differential Geometry of Curves and Surfaces

3. Michael Spivak, Calculus on Manifolds

4. M. Nakahara, Geometry and Physics